Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.

Starting with a fixed resistance of 200 Ω, the necessary values for capacitance and inductance are determined to meet the specifications. First, it is recognized that the damping factor is the reciprocal of twice the product of resistance and capacitance. This insight allows the pinpointing of the required capacitance value for the circuit.

Next, attention is turned to the mathematical expression that connects the resonant frequency, damping factor, and damped natural frequency. By manipulating this equation, resonant frequency information can be extracted.

The resonant frequency, crucial for oscillator design, is inversely proportional to the square root of the product of inductance and capacitance. Armed with this knowledge and the known values, the necessary inductance to meet the design criteria is computed.

These calculations confirm that the damped natural frequency is indeed lower than the resonant frequency, and the inductance value falls below four times the square of the resistance multiplied by capacitance. These conditions affirm that the oscillator circuit has been designed to exhibit the required underdamped oscillations, showcasing proficiency in precise circuit design.